Degree-3 Treewidth Sparsifiers

TL;DR

This paper presents an algorithm to compute degree-3 treewidth sparsifiers as topological minors, maintaining a high treewidth while significantly reducing size and degree, with applications in routing and minor computations.

Contribution

It introduces a polynomial-time algorithm for constructing degree-3 treewidth sparsifiers that preserve treewidth approximately and are small in size, advancing understanding of graph minors and sparsification.

Findings

Produces a topological minor with treewidth Ω(k/polylog(k))

Ensures the minor has O(k^4) vertices

Maintains maximum degree of 3 in the minor

Abstract

We study treewidth sparsifiers. Informally, given a graph $G$ of treewidth $k$, a treewidth sparsifier $H$ is a minor of $G$, whose treewidth is close to $k$, $|V(H)|$ is small, and the maximum vertex degree in $H$ is bounded. Treewidth sparsifiers of degree $3$ are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph $G$ of treewidth $k$, computes a topological minor $H$ of $G$ such that (i) the treewidth of $H$ is $\Omega(k/\text{polylog}(k))$; (ii) $|V(H)| = O(k^4)$; and (iii) the maximum vertex degree in $H$ is $3$. The running time of the algorithm is polynomial in $|V(G)|$ and $k$. Our result is in contrast to the known fact that unless $NP \subseteq coNP/{\sf poly}$, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.